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Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. (English) Zbl 1060.33007
Summary: Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer. Examples such as expansions of generalized hypergeometric functions or Appell functions are discussed. As a further application, we give the general solution of a two-loop integral, the so-called C-topology, in terms of multiple nested sums. In addition, we discuss some important properties of nested sums, in particular we show that they satisfy a Hopf algebra.

MSC:
33B30 Higher logarithm functions
33E20 Other functions defined by series and integrals
33F05 Numerical approximation and evaluation of special functions
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
Software:
FORM ; Nestedsums
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